Is it a big no-no to approach 2 different journalists from the same publication to cover my startup story?

I don't have a lot of experience directly with journalists, but in my experience not really. If one of them was interested but couldn't do it in the first place they would send the piece around the office to see if anyone else wanted it.

At least that's how it's worked with talent agents and entertainers, and they're very similar process'

From my experience I would first contact one and wait a day or two till you get a response -- if any and then proceed to contact another journalist at the same publication. I have found this to be more effective than to contact multiple at the same time.